The Studious Bather

A puzzle from Chris Maslanka’s The Pyrgic Puzzler, 1987:

A bathtub will fill in 3 minutes if the plug is in and the cold tap only is turned on full. It will fill in 4 minutes if the plug is in and the hot tap only is turned on full. With the plug out and both taps off, a full tub will drain in 2 minutes. How long will it take to fill the empty tub if the plug is out and both taps are turned on full?

If the bath’s volume is V, then the cold tap will fill it at a rate of V/3 and the hot tap at a rate of V/4. It will drain at a rate of V/2. Let t be the time it will take to fill it with the plug out and both taps on. Then

(V/3 + V/4 – V/2)t = V

(1/3 + 1/4 – 1/2)t = 1

and t = 12.

(Via Barry R. Clarke’s Mathematical Conundrums, 2023.)